Isometric Composition Operators Research Articles

ISOMETRIES ON SOME GENERAL FAMILY FUNCTION SPACES AMONG COMPOSITION OPERATORS

In this paper, we discuss the isometries of composition operators on the holomorphic general family function spaces F(p, q, s). First, we classify the isometric composition operators acting on a general Banach spaces. For 1 < p < 2, we display that an isometry of Cφ is caused only by a rotation of the disk. We scrutinize the previous work on the case for p ≥ 2. Also, we characterize many of the foregoing results about all α-Besov-type spaces F(p, αp − 2, s), α > 0. We exhibit that in every classes F(p, αp − 2, s) except for the Dirichlet space D = F(2, 0, 0), rotations are the only that produce isometries.

Some results on isometric composition operators on Lipschitz spaces

Given two metric spaces M and N we study, motivated by a question of N. Weaver, conditions under which a composition operator C_phi :{mathrm {Lip}}_0(M)longrightarrow {mathrm {Lip}}_0(N) is an isometry depending on the properties of phi . We obtain a complete characterisation of those operators C_phi in terms of a property of the function phi in the case that B_{{mathcal {F}}(M)} is the closed convex hull of its preserved extreme points. Also, we obtain necessary condition for C_phi being an isometry in the case that M is geodesic.

Isometric composition operators for the Möbius invariant Hp norm of BMOA

Isometric composition operators for the Möbius invariant Hp norm of BMOA

Composition operators on Hardy spaces of the homogenous rooted trees

In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb {T}_{p}\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi \) to be bounded on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces. Also, we discuss the compactness of \(C_\phi \) on \({\mathbb {T}}_{p}\) and finally prove there are no compact composition operators on \({\mathbb {T}}_{p,0}\) spaces.

Operators from the Hardy space to the α‐Bloch space

AbstractLet be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space into the α‐Bloch space on . Also, we show the multiplication operator from into is bounded. Finally, we show that there exist no isometric composition operators.

Isometric composition operators on the Fock-spaces

Isometric composition operators on the Fock-spaces

Isometric composition operators on the Fock-spaces

Isometric composition operators on the Fock-spaces

Weighted Composition Operators from $$H^{\infty }$$ H ∞ to the Bloch Space of Infinite Dimensional Bounded Symmetric Domains

Let \(\mathbb {B}_X\) be a bounded symmetric domain realized as the open unit ball of a JB*-triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space \(H^{\infty }(\mathbb {B}_X)\) into the Bloch space on \(\mathbb {B}_X\). We also give estimates on the operator norm. The lower estimate is an improvement of the result known. We show that the bounded multiplication operators from \(H^{\infty }(\mathbb {B}_X)\) into the Bloch space on \(\mathbb {B}_X\) are precisely those whose symbols are bounded. We also determine the operator norm of the bounded multiplication operator. As a corollary, we show that there are no isometric multiplication operators. Finally, we show that there are no isometric composition operators.

Composition Operators on Weighted Banach Spaces of a Tree

We study composition operators on the weighted Banach spaces of an infinite tree. We characterize the bounded and the compact operators, as well as determine the operator norm and the essential norm. In addition, we study the isometric composition operators.

Isometric composition operators on weighted Dirichlet space

We investigate isometric composition operators on the weighted Dirichlet space \({D_\alpha }\) with standard weights \({(1 - {\left| z \right|^2})^\alpha },\alpha > - 1\). The main technique used comes from Martin and Vukotic who completely characterized the isometric composition operators on the classical Dirichlet space D. We solve some of these but not in general. We also investigate the situation when \({D_\alpha }\) is equipped with another equivalent norm.

Isometric composition operators on the analytic Besov spaces

We investigate the isometric composition operators on the analytic Besov spaces. For 1<p<2 we show that an isometric composition operator is induced only by a rotation of the disk. For p>2, we extend previous work on the subject. Finally, we analyze this same problem for the Besov spaces with an equivalent norm.

Isometric composition operators on weighted Dirichlet-type spaces

Abstract In this article, we characterize the surjective isometric composition operator C φ on the weighted Dirichlet-type spaces of the unit disk D , where φ is an analytic self-map of D , and show that C φ is a surjective isometry if and only if φ is a rotation map. 2000 MSC: Primary: 47B33; Secondary: 30D45; 30H30; 30H05; 47B38.

Determination of Inner Functions by their Value Sets on the Circle

Let Θ be an inner function with finitely many singularities on the unit circle. Then, given distinct points α and β of unit modulus, an explicit expression for Θ is given, in terms of the sets of points on the circle at which Θ takes the values α and β. If Θ has just one singularity, then it is unique to within a hyperbolic automorphism: this property need not hold in general. At the same time, a full characterization of the possible sets at which such values may be taken is obtained. The methods used involve an extension of the Hermite-Biehler Theorem to functions with finitely many essential singularities. Further applications in the theory of isometric composition operators, as well as uniqueness results for restricted shifts involving their numerical ranges are discussed.

Isometric composition operators on the weighted Hardy spaces

AbstractWe investigate the composition operators on the weighted Hardy spaces H2(β). For any bounded weight sequence β, we give necessary conditions for those operators to be isometric. The sufficiency of those conditions is well‐known for the classical space H2. In the case where β is non‐decreasing or non‐increasing, their sufficiency holds only for very few weighted spaces. We find out such spaces by characterizing the isometric monomial composition operators, first for a general β, then for any β as before. With no restriction on β, we provide a complete description of all isometric composition operators. We also prove that the unitary monomial ones are the same as those acting on H2. Such a fact extends to general symbols in the case where β is bounded (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Isometric composition operators on BMOA

AbstractWe characterize and provide examples of the analytic self‐maps of the unit disc, which induce isometric com‐position operators on the space BMOA equipped with a Möbius invariant H2 norm (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Isometric composition operators acting on the Chebyshev space

Much research has been done concerning composition operators over the last five decades. However, this research has primarily focused on a host of standard questions about composition operators in the realm of complex functions. Recently the first author has proposed a number of questions regarding composition operators in the real function area for undergraduate student research supported by the Ursinus College Summer Fellows Program [Doperak 2006; Gareau 2005; Kunaszuk 2006; Potter 2007]. This topic offers accessibility for undergraduate research while providing fertile ground for genuinely new results. This paper focuses on one particular real function space, the Chebyshev space, T , where we explore norm-related ideas for composition operators, specifically norms and isometries. The ultimate goal of this research is to find the norm of a composition operator Cg acting on the Chebyshev space in terms of its symbol g. We begin by considering norms of composition operators in the infinitedimensional space, and move on to examine the topic in the finite-dimensional space. After looking at the various finite-dimensional subspaces, we begin to look at qualities that would lead to the symbol inducing an isometry.

On the isometric composition operators on the Bloch space in Cn

Let $\varphi$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\mathbb{C}^n$. In this work, we show that the composition operator $C_\varphi: f\mapsto f\circ \varphi$ is bounded on the Bloch space $\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $\varphi$ to induce an isometry on $\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_\varphi$ to be an isometry on $\cal{B}$ when $D$ is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric composition operators in the case of the unit disk and for a wide class of symbols on the polydisk.

Preimages of One-Point Sets of Bloch and Normal Functions

We call a sequence {zn} in the open unit disk Δalmost thin if \({\mathop {\lim \sup }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 1,\) where Π(zn) is the product of the pseudo-hyperbolic distances between zn and the other elements in the sequence. We call the sequence thick if \({\mathop {\lim }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 0.\) In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martin and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function φ from Δ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of Δ or if the preimage of each point of Δ contains a sequence along which the hyperbolic derivative of φ approaches one (condition (M’)). In particular, the preimage of each point of Δ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each \(a\in \mathbb{C}\) such that \(f^{-1}(a)=\{z_{n}:n\in \mathbb{N}\}\) is infinite $$ {\mathop {\lim }\limits_{n \to \infty } }\left(1 - |z_{n} |^{2} \right)|f'{\left( {z_{n} } \right)}| = 0. $$ Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of Δ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in Δ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk Δn to be the symbol of an isometric composition operator on the Bloch space of Δn and discuss a higher-dimensional analogue of condition (M’).

Characterisation of the isometric composition operators on the Bloch space

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

INVERTIBLE AND ISOMETRIC COMPOSITION OPERATORS ON VECTOR-VALUED HARDY SPACES

Invertible and isometric composition operators acting on vector-valued Hardy space (E) are characterized.